tppreqb - Metamath Proof Explorer

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Theorem tppreqb 4174

Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)

**Assertion**
Ref	Expression
tppreqb	$/\$ $/\$

**Proof of Theorem tppreqb**
Step	Hyp	Ref	Expression
1		3ianor 990	. . . 4 $/\$ $/\$ $\/$ $\/$
2		df-3or 974	. . . 4 $\/$ $\/$ $\/$ $\/$
3	1, 2	bitri 249	. . 3 $/\$ $/\$ $\/$ $\/$
4		orass 524	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
5		ianor 488	. . . . . . . 8 $/\$ $\/$
6		tpprceq3 4173	. . . . . . . 8 $/\$
7	5, 6	sylbir 213	. . . . . . 7 $\/$
8		tpcoma 4129	. . . . . . 7
9		prcom 4111	. . . . . . 7
10	7, 8, 9	3eqtr3g 2531	. . . . . 6 $\/$
11		orcom 387	. . . . . . . 8 $\/$ $\/$
12		ianor 488	. . . . . . . 8 $/\$ $\/$
13	11, 12	bitr4i 252	. . . . . . 7 $\/$ $/\$
14		tpprceq3 4173	. . . . . . 7 $/\$
15	13, 14	sylbi 195	. . . . . 6 $\/$
16	10, 15	jaoi 379	. . . . 5 $\/$ $\/$ $\/$
17	4, 16	sylbi 195	. . . 4 $\/$ $\/$ $\/$
18	17	orcs 394	. . 3 $\/$ $\/$
19	3, 18	sylbi 195	. 2 $/\$ $/\$
20		df-tp 4038	. . . 4
21	20	eqeq1i 2474	. . 3
22		ssequn2 3682	. . . 4
23		snssg 4166	. . . . . . 7
24		elpri 4053	. . . . . . . 8 $\/$
25		nne 2668	. . . . . . . . . 10
26		3mix2 1166	. . . . . . . . . 10 $\/$ $\/$
27	25, 26	sylbir 213	. . . . . . . . 9 $\/$ $\/$
28		nne 2668	. . . . . . . . . 10
29		3mix3 1167	. . . . . . . . . 10 $\/$ $\/$
30	28, 29	sylbir 213	. . . . . . . . 9 $\/$ $\/$
31	27, 30	jaoi 379	. . . . . . . 8 $\/$ $\/$ $\/$
32	24, 31	syl 16	. . . . . . 7 $\/$ $\/$
33	23, 32	syl6bir 229	. . . . . 6 $\/$ $\/$
34		3mix1 1165	. . . . . . 7 $\/$ $\/$
35	34	a1d 25	. . . . . 6 $\/$ $\/$
36	33, 35	pm2.61i 164	. . . . 5 $\/$ $\/$
37	36, 1	sylibr 212	. . . 4 $/\$ $/\$
38	22, 37	sylbir 213	. . 3 $/\$ $/\$
39	21, 38	sylbi 195	. 2 $/\$ $/\$
40	19, 39	impbii 188	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $\/$ wo 368 $/\$ wa 369 $\/$ w3o 972 $/\$ w3a 973

wceq 1379

wcel 1767

wne 2662

cvv 3118

cun 3479

wss 3481

csn 4033

cpr 4035

ctp 4037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1382 df-ex 1597 df-nf 1600 df-sb 1712 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-v 3120 df-dif 3484 df-un 3486 df-in 3488 df-ss 3495 df-nul 3791 df-sn 4034 df-pr 4036 df-tp 4038

This theorem is referenced by: (None)

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